5th presentation: Fuzzy presentation addition of normalization method
1.0 Determine the alternatives and criteria
k decisions makers Dt, t=1~k
m alternatives Ai, i=1~m
n criteria Cj, j=1~n
We assume that the importance weights of criteria and the rating performance ratings
under each of the qualitative criterias are assessed in linguistic term represented by positive
trapezoidal fuzzy number.
The competitiveness will be determined upon their final ranking values. The criteria
could be both quantitative and quantitative in order to integrate every aspect in the evaluation.
2.0 Aggregate the importance weights
In this stage, decision-makers start to assess the degree of importance of each criterion
and compute the aggregated weights of individual criterion
Let
wjt = ( ajt,bjt,cjt,djt), wjt R+, j=1,2,….,n, t=1,2,….,k
Be the importance weight given by decision maker Dt to criterion Cj
And
(4.1)
Where
Be the degree of importance weight, the degree of importance weight is quantified by
linguistic terms represented by fuzzy numbers.
LINGUISITIC VALUES
(SI) Slightly Importance
(FI) Fairly Importance
(I) Importance
(VI) Very Importance
(EI) Extremely Importance
FUZZY NUMBERS
(0,0,0.1,0.3)
(0,0.2,0.3,0.5)
(0.3,0.45,0.55,0.7)
(0.5,0.7,0.8,1)
(0.7,0.9,1,1)
3.0 Aggregate the rating of alternative versus criteria
3.1 To evaluate the alternatives under each of the criteria
Let
Be the linguistic rating assigned to Alternative
by decision-maker
criterion
assessed by the committee of k decision-makers can be evaluated as
under
Where
3.2 To evaluate the alternatives versus Quantitative Criteria
For making the comparison among the variety of criteria, we have to transform the
incomparable quantitative data ,
, into the normalized value, . In this paper we use
“Normal distribution” concept in normalizing the quantitative data.
•
Benefit criteria: Cj, j=(e+1)~f The larger the better
Normalization equations:
•
Cost criteria: Cj, j=(f+1)~n The smaller the better
Normalization equations:
Note:
= Student's t-score for alternative
in criteria
= Non-normalized quantitative data of alternative
= Standard deviation of Criteria
in criteria
from alternative
= The mean of non-normalized quantitative data from
alternative
3.3 To evaluate the alternatives versus Qualitative Criteria
Qualitative Criteria Cj, j=1~k are measured by linguistic values represent by fuzzy
number
4.0 Develop Membership Function for FWA using Interval Analysis and
4.1 Making Final fuzzy equation for Final fuzzy value,
Fuzzy weight average equation:
(4.5)
Deriving Fuzzy weight average equations
and
are derived as follows:
(4.6)
(4.7)
Then equations (4.6) and (4.7) are applied to (4.5) to obtain:
Where:
(4.8)
(4.9)
(4.10)
(4.11)
Denoted that:
Equation (4.11) and (4.12) can be express as:
(4.13)
(4.14)
Only roots in [0, 1] will be retain in (4.12) and (4.13). The left and the right membership
functions can be developed as:
(4.15)
Where:
For convenience, the
can be expressed as:
4.2 Ranking Obtain
The final fuzzy evaluation value, , the larger
the higher priority the alternative
will have. The subtraction of the left relative area from the right relative area
the ranking function.
Therefore, for any two fuzzy numbers
is used as
and , if:
That is, the smaller the defuzzification value
will have.
Where;
the higher priority the alternative
In order to make the calculation easier and more convenient, we applied “Inverse
functions concept” in the ranking function